The present invention relates to a tennis racket, and more particularly relates to improvement in spin performance of the face of a tennis racket at shooting balls.
In general construction of a tennis racket, a substantially oval ring shaped head frame defines a face constructed by a string network and the string network is made up of interlaced main (longitudinal) and cross (transverse) strings mounted under tension to the head frame. The main strings are usually kept under a tension in a range from 26 to 30 kg and the string tension ratio (T.sub.1 /T.sub.2), i.e. the ratio of the main string tension (T.sub.1) to the cross string tension (T.sub.2), is set to a value in a range from 1/1 to 2/1. By setting the string tension ratio to a value in this range, the main and cross string tensions well balance so that the head frame after string setting should preserve its original shape before string setting.
Among various performances of a tennis racket at shooting ball, high degree of spin performance is required by players, in particular by high level players. Here, the term "spin performance" refers to an operation of a racket face to rotate a ball in a direction intended by a player at shooting. For example, top spin causes intensive forward rotation of a ball and back spin causes intensive rearward rotation of a ball.
It is well known in the field of art that spin efficiency, i.e. the degree of spin performance on a ball, is dependent upon the magnitude of the friction force acting on the ball from the face at the very moment of collision. It is also confirmed that, with the above-described construction of a racket face, about one half of the normal reaction acting on a ball at shooting is lost without any contribution to its friction force. Here, the term "normal reaction" refers to a reactive force acting on a ball in a direction normal to the racket face shooting the ball. In order to increase the degree of such a contribution, it is helpful to increase the value of the above-described main/cross string tension ratio (T.sub.1 /T.sub.2).
Now, the value of compressive rigidity of a head frame is in a range from 12 to 18 Kgf/mm when measured in the direction of main strings. For this measurement, a tennis racket is fixed at the heel of its grip and a load of 10 Kg is applied to the crown of its head frame.
As stated already, the main/cross string tension ratio is conventionally set to a value in a range from 1/1 to 2/1 for stable balance between main and cross string tensions. When the string tension ratio exceeds this limit, unduly increased main string tension would cause longitudinal compression and lateral expansion of the head frame. Such deformation in excess tends to cause breakage of the head frame. Even when no serious breakage is caused, such deformation causes undesirable disorder in main/cross tension balance on the racket face.
Regarding the mechanism of the above-described spin performance of a racket face, it was confirmed by the inventor of the present invention that the degree of spin performance is closely related to dynamic behaviour of a ball and a racket face at mutual collision. More specifically, the most important factor in spin performance is created by the correlationship between the main/cross string tension ratio (T.sub.1 /T.sub.2) and the mode of distribution of normal reaction, i.e. normal reactive force, from the racket face.
The values of main and cross string normal reactions are given as follows. It is here assumed that a ball is shot at an intersection of a main string with a cross string in a racket face. Then, the normal reaction (N.sub.1) of the main string is given by; EQU N.sub.1 =(4T.sub.1 /L.sub.1).multidot.X.sub.1 ( 1)
L.sub.1 : length of the main string PA1 X.sub.1 : displacement of the main string in the normal direction. PA1 L.sub.2 : length of the cross string PA1 X.sub.2 : displacement of the cross string in the normal direction. PA1 F.sub.1 : frictional force from the main strings PA1 F.sub.2 : frictional force form the cross strings PA1 T: string tension PA1 L: length of the string concerned PA1 x: displacement of the string in the normal direction.
Whereas, the normal reaction (N.sub.2) of the cross string is given by; EQU N.sub.2 =(4T.sub.2 /L.sub.2).multidot.X.sub.2 ( 2)
The total normal reaction (N) acting on the ball is then given by; EQU N=N.sub.1 +N.sub.2 ( 3)
In the construction of a conventional tennis racket, its racket face is designed to suffice the following relationship; EQU L.sub.2 /L.sub.1 .apprxeq.T.sub.2 /T.sub.1 ( 4)
From this equation, the following relationship is deduced; EQU T.sub.2 /L.sub.2 =T.sub.1 /L.sub.1 =constant (5)
This equation endorses an inference that the normal reaction (N.sub.2) from the cross strings is roughly equal in amount to the normal reaction (N.sub.1) from the main strings. This inference is believed to be safely propagated to the entire area of a racket face and the total reaction acting on a ball at collision is almost equally shared by its main and cross strings.
When striking of a ball against a racket face is microscopically analyzed as a mechanical model, the general collision consists of its impact contact with main strings and its impact contact with cross strings. At these impact contacts, a frictional force acts on the ball from the face and this frictional force (F) is given by; EQU F=F.sub.1 +F.sub.2 ( 6)
Then, when the above-described normal reactions N.sub.1 and N.sub.2 are taken into consideration, these values are given by; EQU F.sub.1 =.mu..sub.1 N.sub.1 ( 7) EQU F.sub.2 =.mu..sub.2 N.sub.2 ( 8) EQU .thrfore.F=.mu..sub.1 N.sub.1 +.mu..sub.2 N.sub.2 ( 9)
Here, .mu..sub.1 indicates the dynamic friction coefficient between the ball and the main strings in the lateral direction of the latter whereas .mu..sub.2 indicates the dynamic friction coefficient between the ball and the cross strings in the longitudinal direction of the latter.
When attention is directed to one string in a racket face, its dynamic friction coefficient in the lateral direction is apparently far greater than its dynamic friction coefficient in the longitudinal direction. Taking into consideration the fact that, in construction of a common racket face, its main strings and cross strings are usually made of a same material and that, as a consequence, same in physical properties, this relationship between the lateral and longitudinal dynamic friction coefficients can be safely applied to the relationship of the above-described equation (9).
Thus, when compared with the degree of influence of the normal reaction (N.sub.1) of the main strings on the total frictional force (F) acting on the ball, the degree of influence of the normal reaction (N.sub.2) of the cross strings is quite small. In the case of the conventional racket face, the normal reaction (N.sub.2) from the cross strings roughly equals in amount the normal reaction (N.sub.1) from the main strings as inferred on the basis of the above-described equation (5). Stated otherwise, as briefed already, about half of the total normal reaction (N) is wasted without any contribution to creation of the frictional force which is useful for raising spin performance of the racket face.
On the basis of the foregoing analysis, it was first intended by the inventor of the present invention to increase the frictional force (F) acting on a ball from a racket face by means of raising the ratio (N.sub.1 /N.sub.2) of the normal reaction (N.sub.1) of the main strings to the normal reaction (N.sub.2) of the cross strings. Rise in this ratio (N.sub.1 /N.sub.2) satisfies the following relationship; EQU N.sub.1 /N.sub.2 &gt;1 (10)
Here, the above-described increase in frictional force (F) intended by the inventor is resulted from a combination of the relationship in dynamic friction coefficient (.mu..sub.1 &gt;.mu..sub.2) with the relationship in normal reaction (N.sub.1 &gt;N.sub.2).
From the equations (1) and (2), the normal reaction (N) of a string is generally given by; EQU N=(4T/L).multidot.x (11)
As is clear from this relationship, the magnitude of the normal reaction (N) is proportional to the magnitude of the string tension (T). Consequently, rise in normal reaction ratio (N.sub.1 /N.sub.2) can be achieved by rise in string tension ratio (T.sub.1 /T.sub.2). In other words, the larger the string tension ratio (T.sub.1 /T.sub.2), the larger the normal reaction ratio (N.sub.1 /N.sub.2).
As stated above, the conventional tennis racket is generally designed so that the value of compressive rigidity of its head frame is in a range from 12 to 18 Kgf/mm when measured in the direction of its main strings. When the string tension ratio (T.sub.1 /T.sub.2) is increased carelessly, resultant main string tension would be increased to cause longitudinal compression and lateral expansion of the head frame. As stated already, such deformation in excess is liable to cause breakage of the head frame or serious disorder in main/cross tension balance on the racket frame.